Sectoral versus Aggregate Shocks: A Structural
Factor Analysis of Industrial Production
Andrew T. Foerster
Duke University
Pierre-Daniel G. Sarte
Federal Reserve Bank of Richmond
Mark W. Watson
Princeton University and National Bureau of Economic Research
Using factor methods, we decompose industrial production (IP) into
components arising from aggregate and sector-specific shocks. An approximate factor model finds that nearly all of IP variability is associated with common factors. We then use a multisector growth model
to adjust for the effects of input-output linkages in the factor analysis.
Thus, a structural factor analysis indicates that the Great Moderation
was characterized by a fall in the importance of aggregate shocks while
the volatility of sectoral shocks was essentially unchanged. Consequently, the role of idiosyncratic shocks increased considerably after
the mid-1980s, explaining half of the quarterly variation in IP.
We are grateful to an anonymous referee and two editors for very useful comments.
We are especially indebted to Andreas Hornstein for helpful suggestions, and we thank
Robert King, Ray Owens, Roy Webb, and seminar participants at various conferences and
universities for helpful conversations and comments. Support was provided by the National
Science Foundation through grant SES-0617811. The views expressed in this paper are
those of the authors and do not necessarily reflect those of the Federal Reserve Bank of
Richmond or the Federal Reserve System. Data and replication files for this research can
be found at http://www.princeton.edu/∼mwatson.
[Journal of Political Economy, 2011, vol. 119, no. 1]
䉷 2011 by The University of Chicago. All rights reserved. 0022-3808/2011/11901-0001$10.00
1
2
I.
journal of political economy
Introduction
The Federal Reserve Board’s Index of Industrial Production (IP) is an
important indicator of aggregate economic activity in the United States.
Month-to-month and quarter-to-quarter variations in the index are large.
Monthly and quarterly growth rates for the seasonally adjusted IP index
over 1972–2007 are plotted in figure 1. Over this sample period, the
standard deviation of monthly growth rates was over 8 percentage points
(at an annual rate), and quarterly growth rates had a standard deviation
of nearly 6 percentage points. Also evident in the figure is the large fall
in volatility associated with the Great Moderation; the standard deviation
in the 1984–2007 period is roughly half its pre-1984 value for both the
monthly and quarterly series.
Because the IP index is constructed as a weighted average of production indices across many sectors, the large volatility in the IP index
is somewhat puzzling. Simply put, while production in an individual
sector (e.g., motor vehicle parts) may vary substantially from month to
month or quarter to quarter, apparently much of this variability does
not “average out” in the index of economywide production. There are
three leading explanations for this puzzle. The first relies on aggregate
shocks that affect all industrial sectors. Since these shocks are common
Fig. 1.—Growth rates of industrial production. Monthly: thin line; quarterly: thick line.
Percentage points are at annual rates.
sectoral versus aggregate shocks
3
across sectors, they do not average out and thus become the dominant
source of variation in aggregate economic activity. The other two explanations rely on uncorrelated sector-specific shocks. The first of these,
carefully articulated in Gabaix (2011), postulates a handful of very large
sectors in the economy, and because these large sectors receive a large
weight in aggregate IP, shocks to these sectors do not average out.
The second sector-specific explanation postulates complementarities
in production, such as input-output linkages, that propagate sectorspecific shocks throughout the economy in a way that generates substantial aggregate variability. The purpose of this paper is to determine
the quantitative importance of these three explanations for explaining
the variability in aggregate U.S. industrial activity as measured by the
IP index.
The related literature that analyzes sector-specific versus aggregate
sources of variations in the business cycle has followed two main approaches. Long and Plosser (1987), Forni and Reichlin (1998), and
Shea (2002), among others, rely on factor analytic methods, coupled
with broad identifying restrictions, to assess the relative contributions
of aggregate and sector-specific shocks to aggregate variability. These
papers generally find that sector-specific shocks contribute a nontrivial
fraction of aggregate fluctuations (e.g., approximately 50 percent in
Long and Plosser [1987]). A second strand of literature is rooted in
more structural calibrated multisector models, such as Long and Plosser
(1983), Horvath (1998, 2000), Dupor (1999), and Carvalho (2007), that
explicitly take into account sectoral linkages across sectors. In these
models, whether sectoral linkages are sufficiently strong to generate
substantial aggregate variability from idiosyncratic shocks depends in
part on the exact structure of these linkages.
This paper bridges these reduced-form and structural approaches and
sorts through the leading explanations underlying both the volatility of
IP and its decline in the Great Moderation period. In particular, it
describes conditions under which neoclassical multisector models that
explicitly consider sectoral linkages, such as those of Long and Plosser
(1983) or Horvath (1998), produce an approximate factor model as a
reduced form. Aggregate shocks to sectoral productivity emerge as the
common output factors in the approximate factor model. The “uniquenesses” in the factor model are associated with sector-specific shocks.
However, because sectoral linkages induce correlation across the uniquenesses, factors estimated from reduced-form models may be biased and
reflect not only aggregate shocks but also idiosyncratic shocks that propagate across sectors by way of these sectoral linkages.
We begin our analysis in Section II by describing data on 117 sectors
that make up aggregate IP and that form the basis of our empirical
analysis. The reduced-form analysis that we carry out in this section
4
journal of political economy
suggests that, in contrast to Gabaix (2011), sectoral weights play little
role in explaining the variability of the aggregate IP index. As in Shea
(2002), aggregate variability is driven mainly by covariability across sectors and not sector-specific variability. This leads us to rule out the “fewlarge-sectors” explanation in favor of explanations that rely on covariability across sectors.
In Section III, we focus on this covariability. Consistent with Quah
and Sargent (1993), who study comovement in employment across 60
industries, and Forni and Reichlin (1998), who study annual U.S. output
and productivity data over 1958–86, we find that much of the covariability in sectoral production can be explained by a small number of
common factors. These common factors are the leading source of variation in the IP index, and a decrease in the variability of these common
factors explains virtually all of the 1984–2007 decline in aggregate volatility.
Because common factors may reflect not only aggregate shocks but
also the propagation of idiosyncratic shocks by way of sectoral linkages,
Section IV develops a generalized version of the multisector growth
model introduced by Long and Plosser (1983), and later extended by
Horvath (1998), to filter out the effects of these linkages. The generalized model we present is detailed enough to accommodate our decomposition of IP into 117 sectors while still allowing these sectors to
interact via input-output linkages in both intermediate materials and
capital goods. Empirical results from the model suggest that the onset
of the Great Moderation is characterized by a fall in the importance of
aggregate shocks. In contrast, the volatility of sectoral shocks remained
approximately unchanged throughout the 1972–2007 sample period.
As a result, sectoral shocks acquired a more prominent role in the latter
part of our sample period. More specifically, the model implies that
while sector-specific shocks accounted for 20 percent of the variability
in aggregate IP during 1972–83, this share increased to roughly 50
percent during 1984–2007.
The model developed in Section IV is sufficiently complicated that
we must solve parts of it using numerical methods, and this hides some
of the key mechanisms that are responsible for its empirical predictions.
Hence, in Section V, we use insights obtained from analytic solutions
to the models of Long and Plosser (1983), Horvath (1998), Dupor
(1999), and Carvalho (2007), together with numerical experiments, to
elucidate these mechanisms. In particular, we show that the model’s
predictions depend on the nature of contemporaneous input-output
interactions associated with intermediate materials as well as intertemporal input-output interactions associated with capital goods. For example, in the original work of Long and Plosser (1983), the structural
model imposes a one-period lag in the delivery of materials and abstracts
sectoral versus aggregate shocks
5
from capital accumulation. Consequently, when confronted to U.S. data
using factor analytic methods, that model predicts a relatively limited
role for idiosyncratic sectoral shocks. In contrast, the propagation of
sectoral shocks is enhanced when materials are used contemporaneously
in production and capital goods in a sector are potentially produced in
other sectors.
Section VI carries out some robustness analysis and shows, for example, that the empirical conclusions in Section III are robust to
changes in sectoral input-output relationships over our 35-year sample
period. Section VII offers some concluding remarks.
II.
A First Look at the Sectoral IP Data
A.
Overview of the Data
This paper uses sectoral data on IP over the period 1972–2007. The
data are for 117 sectors that make up aggregate IP, where each sector
roughly corresponds to a four-digit industry using the North American
Industry Classification System (NAICS). As discussed in Appendix A,
where the data are described in detail, this is the finest level of disaggregation that allows us to match the sectors to the input-output and
capital use tables used to calibrate the structural models in Section IV.
Let IPt denote the value of the aggregate IP index at date t, and IPit
denote the index for the ith sector. These indices are available monthly.
Quarterly values of the indices are constructed as averages of the months
in the quarter. Figure 1 indicates that the quarterly and monthly data
share many features, except that monthly data exhibit more highfrequency variability. Because we are less interested in this highfrequency variability, we will focus our attention on the quarterly data.
Growth rates (in percentage points) are denoted by g t for aggregate IP
and by x it for sectoral IP, where g t p 400 # ln (IP/IP
t
t⫺1), and similarly
for x it . We will let N denote the number of sectors in our sample (so
that N p 117 sectors) and T denote the number of time periods (T p
144 quarters).
The sectoral growth rates are more volatile than the aggregate growth
rates plotted in figure 1. Figure 2 shows the distributions of standard
deviations of growth rates for the 117 IP sectors over the sample periods
1972–83 and 1984–2007. The Great Moderation is evident in these distributions, where the median sectoral growth rate standard deviation
fell from 17.4 in the early sample period to 11.3 in the latter period.
Fig. 2.—Standard deviation of sectoral growth rates; percentage points are at annual
rates. A, 1972–83; B, 1984–2007.
sectoral versus aggregate shocks
B.
7
Sectoral Size, Covariability, and Aggregate Variability
Let wit denote the share of the ith sector in the aggregate index, so that
N
the growth rate of aggregate IP can be written as g t p 冘ip1 witx it . This
subsection investigates the role of the sectoral weights, wit , and that of
the covariance of x it across sectors in explaining the variability in g t .
Our focus on the role of wit is motivated by recent work by Gabaix
(2011), who considers an economy in which N firms operate independently. The growth rate in aggregate output is then a share-weighted
average of the firms’ growth rates, which are assumed to be mutually
uncorrelated. Gabaix derives conditions on the distribution of firm size
that guarantee that the variability of the aggregate growth rate remains
bounded away from zero as the number of firms grows large. The result
relies on a size distribution that places sufficient weight on relatively
large firms so that shocks to these firms have a nonnegligible effect on
aggregate output.
In our context, Gabaix’s analysis suggests that the volatility of aggregate IP may arise from idiosyncratic shocks to IP sectors with large shares.
In our data, the empirical distribution of the 117 share weights suggests
that this could indeed be an important source of variability in aggregate
IP: with 117 sectors, the average share is less than 1 percent, but the
four largest sectors account for 20 percent of aggregate IP, and the 20
largest sectors account for more than 50 percent. Thus, the volatility of
aggregate IP may arise from shocks to a relatively small number of sectors
that receive a large weight in the aggregate average.
To quantify the share effect, consider the following decomposition of
g t:
冘
N
gt p
冘 冘(
N
witx it p
ip1
N
)
1
1
x it ⫹
wit ⫺
x it,
N ip1
N
ip1
(1)
where the first equality defines the aggregate growth rate and the second
decomposes g t into a component associated with equal shares 1/N and
a component associated with the deviation of shares from 1/N.1 The
N
first term, (1/N ) 冘ip1 x it , weights each sector equally, and when x it are
uncorrelated, this component has a variance proportional to N ⫺1 and
so will be small when N is large. However, the variance of the second
N
term, 冘ip1 [wit ⫺ (1/N )]x it , may be large if the cross-sectional variance of
T
shares is large at date t. When we let w̄i p (1/T ) 冘tp1 wit denote the
average share of sector i, this second term may be large because some
sectors have large average shares over the sample period (so w̄i is large
for some i) or have very unstable shares (so for each date t, the value
¯ i is large for some sector i). To capture these two distinct effects,
of wit ⫺ w
it useful to further decompose g t as
1
Gabaix (2011) refers to the term
冘
N
ip1
[wit ⫺ (1/N)]xit as the “granular residual.”
8
冘 冘(
N
gt p
N
journal of political economy
) 冘
N
1
1
¯i ⫺
¯ i)x it.
x it ⫹
w
x ⫹ (w ⫺ w
N ip1
N it ip1 it
ip1
(2)
Figure 3 plots this decomposition of g t , and table 1 shows the standard
deviation of each of the components over the full sample and the two
subsamples. Examination of the figure and table indicates that the
N
equally weighted component, N ⫺1 冘ip1 x it , tracks the aggregate IP series
closely. (Indeed, from table 1, this component has a variance slightly
larger than the aggregate, g t .) In contrast, the unequal share components are far less important. Therefore, this decomposition leads us to
conclude that although aggregate IP is composed in part of a few large
sectors, the mechanism highlighted by Gabaix (2011) is not an important factor in explaining the variability of aggregate IP.
While the distribution of sectoral shares (i.e., differences in the size
of sectors) is not a crucial consideration for the variability of IP, the
covariability of sectoral growth rates is an important factor in this calculation. Tables 2 and 3 summarize key results to clarify this point.
Table 2 shows that the average pairwise correlation of the sectoral
growth rates was positive and large (0.27) in the 1972–83 sample period
and fell substantially (to 0.11) in the 1984–2007 sample period. Table
3 summarizes a calculation suggested by Shea (2002). In particular,
Fig. 3.—Share weight decomposition of industrial production; percentage points are
at annual rates.
sectoral versus aggregate shocks
9
TABLE 1
Share Weight Decomposition of Industrial Production
Series
1972–2007
1972–83
1984–2007
5.8
8.7
3.6
6.9
2.0
.8
10.5
2.8
1.0
4.2
1.5
.6
冘
冘
gt p witxit
components
(1/N) xit
¯ i ⫺ (1/N)]xit
[w
¯ i)xit
(wit ⫺ w
冘
冘
Note.—Entries are the sample standard deviations of the growth rate of
IP, gt , and its components. Percentage points are at annual rates.
TABLE 2
Average Pairwise Correlations of Sectoral
Growth Rates
1972–2007
.19
1972–83
1984–2007
.27
.11
Note.—Entries are the sample correlations of
xit and xjt averaged over all i ( j.
TABLE 3
Standard Deviation of gt With and Without Sectoral Covariance
1972–2007
1972–83
1984–2007
A. Using Actual (wit) Share Weights
With sectoral covariation
Without sectoral covariation
5.8
1.9
With sectoral covariation
Without sectoral covariation
6.9
1.8
8.7
2.4
3.6
1.6
B. Using Equal (1/N) Share Weights
10.5
2.5
4.2
1.4
Note.—The entries for rows labeled “with sectoral covariation” are the sample standard
deviations of i witxit (panel A) and N⫺1 i xit (panel B). The entries labeled “without sectoral
covariation” are computed as
冘
冘
冑T⫺1 冘t 冘i hit2(xit ⫺ x¯i)2,
where hit p wit in panel A and hit p N⫺1 in panel B. Percentage points are at annual rates.
because g t is a sum (p冘ip1 witx it), the variance of g t is the sum of the
variances of the summands plus all of the pairwise covariances. Thus,
table 2 compares the standard deviation of g t computed using the usual
formula (which includes all of the covariances) to the value from the
formula that considers only the individual variances of x it . The results
indicate that the covariance terms are the dominant source of variation
in the growth rate of aggregate IP. For example, in panel A of table 3,
the full-sample standard deviation of g t falls by a factor of three (from
5.8 percent to 1.9 percent) when the covariance terms are set equal to
N
10
journal of political economy
zero. This result also holds for a version of aggregate IP that weights
each sector equally and therefore is not driven by a relatively few covariances associated with large sectors (panel B).
In summary, the results in this subsection allow us to discount the
“large shocks in sectors with large shares” explanation for the variability
of IP growth. If anything, the fact that, in table 3, the standard deviation
of IP growth using uniform 1/N shares is slightly larger than that computed using actual share weights suggests that larger sectors are associated with smaller shocks. In addition, the variability of aggregate IP
growth is associated with shocks that lead to covariability in sectoral
output, not shocks that lead to large idiosyncratic sectoral volatility.
Therefore, the remaining challenge is to measure and understand the
shocks that lead to covariability, and this challenge is taken up in the
remainder of the paper.
III.
Statistical Factor Analysis
As discussed in Forni and Reichlin (1998), the approximate factor model
is a natural way to model the covariance matrix of sectoral production.
With X t representing the N#1 vector of sectoral growth rates, this model
represents X t as
X t p LFt ⫹ u t,
(3)
where Ft is a k#1 vector of latent factors, L is an N#k matrix of coefficients called factor loadings, and u t is an N#1 vector of sector-specific
idiosyncratic disturbances. In classical factor analysis (Anderson 1984),
Ft and u t are mutually uncorrelated independent and identically distributed (i.i.d.) sequences of random variables, and u t has a diagonal
covariance matrix. Thus, X t is an i.i.d. sequence of random variables
with covariance matrix SXX p LSFF L ⫹ Suu, where SFF and Suu are the
covariance matrices of Ft and u t , respectively. Because Suu is diagonal,
any covariance between the elements of X t arises from the common
factors Ft .
Approximate factor models (see Chamberlain and Rothschild 1983;
Connor and Korackzyk 1986; Forni et al. 2000; Stock and Watson 2000)
weaken the classical factor model assumptions by (essentially) requiring
that, in large samples, key sample moments involving Ft and u t mimic
the behavior of sample moments in classical factor analysis. This allows
for weak cross-sectional and temporal dependence in the series, subject
to the constraint that sample averages satisfy laws of large numbers with
the same limits as those that would obtain in classical factor analysis.
When N and T are large, as they are in this paper’s application, the
approximate factor model has proven useful because relatively simple
methods can be used for estimation and inference. For example, pe-
sectoral versus aggregate shocks
11
TABLE 4
Decomposition of Variance from the Statistical Two-Factor
Model
Standard deviation (data)
Standard deviation (model)
R 2(F)
1972–83
1984–2007
8.7
8.5
.89
3.6
3.6
.87
Note.—The first row shows the sample standard deviation of gt , and
the second row shows the implied standard deviation from the factor
model with constant shares and uncorrelated uit . The final row shows
the fraction of the variance of gt accounted for by the common factors.
nalized least-squares criteria can be used to consistently estimate the
number of factors (Bai and Ng 2002), principal components can be
used to consistently estimate factors (Stock and Watson 2000), and the
estimation error in the estimated factors is sufficiently small that it can
be ignored when estimating variance decompositions or conducting
inference about L (Stock and Watson 2000; Bai 2003).
Tables 4 and 5 and figures 4 and 5 summarize the results from applying these methods to the sectoral IP growth rates. To begin, we estimated the number of factors using the Bai and Ng (2002) information
criteria for principal components (their ICP1 and ICP2 criteria). These
estimators yielded two factors in the full sample period (1972–2007)
and the first sample period (1972–83). They yielded one factor in the
second sample period (1984–2007). The results shown in tables 4 and
5 and figures 4 and 5 are based on estimated two-factor models. For
robustness, we also carried out our analysis using one- and three-factor
models. The results (not shown) were similar to those we report for the
two-factor model.
We gauge the importance of common shocks, F, relative to the idiosyncratic shocks, u, in two related ways. First, we calculate the fraction
of variability in the growth rates of aggregate IP explained by F. Second,
we calculate the fraction of the variability in each sector’s growth rate
that is explained by F. The fraction of variability in x it associated with
the common factors is given by R i2(F ) p li SFF l i /ji2, where li denotes
the ith row of L and ji2 p Var (x it). To compute the analogous R 2 for
aggregate IP, we ignore time variation in wit and use the approximation
¯ X t , where w
¯ is the vector of mean sectoral shares. The fraction
gt ≈ w
of the variability of g t attributable to the factors is then R 2(F ) p
¯ LSFF Lw/j
¯ g2.
w
Table 4 shows the two-factor model’s implied standard deviation of
aggregate IP (computed using constant shares, w̄) together with
R 2(F ). The factor model implies an aggregate IP index with volatility
that is essentially identical to that found in the data, and the common
12
journal of political economy
TABLE 5
Fraction of Variability in Selected Sectoral Growth Rates
Explained by Common Factors
Sector
A. 1972–83:
Other fabricated metal products
Fabricated metals: forging and stamping
Machine shops: turned products and screws
Commercial and service industry machinery/other
general purpose machinery
Foundries
Other electrical equipment
Metal working machinery
Fabricated metals: cutlery and handtools
Electrical equipment
Architectural and structural metal products
B. 1984–2007:
Coating, engraving, heat treating, and allied activities
Plastic products
Commercial and service industry machinery/other
general purpose machinery
Fabricated metals: forging and stamping
Household and institutional furniture and kitchen cabinets
Veneer, plywood, and engineered wood products
Metal working machinery
Foundries
Millwork
Other fabricated metal products
Ri2(F)
.86
.85
.83
.83
.80
.79
.78
.76
.73
.72
.68
.67
.65
.65
.59
.59
.52
.52
.51
.50
Note.—The entries are the sectors ordered by Ri2(F) p liSFFli/j2i , where
parameters and moments are estimated over the periods shown.
factors explain nearly all of the variability in growth rates of the aggregate IP index over both sample periods. Because the common factors
explain roughly 90 percent of the variability in both sample periods,
they are responsible for 90 percent of the decrease in aggregate volatility
across the two time periods. Figure 4 plots the growth rate of aggregate
ˆ ˆ ). The fitted value
IP and its fitted value from the factor model (w̄ LF
t
closely tracks the actual value during the entire sample period. Thus,
viewed through the lens of the statistical factor model, the Great Moderation is explained by a decrease in the variance of the common shocks
that affect IP.
At a disaggregated level, figure 5 shows the distribution R i2(F ) for the
two sample periods. Common shocks were an important source of variability in sectoral growth rates prior to 1984 (the median value of
R i2(F ) is 0.41) but were less important after 1984 (the median value of
R i2(F ) falls to 0.19). While the importance of common shocks for the
variability of sectoral growth rates declines after 1984, figure 5 indicates
sectoral versus aggregate shocks
13
Fig. 4.—Factor decomposition of industrial production; percentage points are at annual
rates.
that these shocks nevertheless explain a large fraction of output growth
volatility in several individual series in both sample periods. Table 5 lists
the 10 sectors with the largest fraction of variability accounted for by
common shocks. Prior to 1984, for example, idiosyncratic shocks played
virtually no role in the variability of output growth in the sectors related
to fabricated metals: forging and stamping or other fabricated metal
products.
Because the sectors in table 5 move mainly with common shocks and
movements in the aggregate IP index are associated with these shocks,
the sectors listed in table 5 turn out to be particularly informative about
the IP index. Consider, for instance, the problem of tracking movements
in IP in real time using only a subset M of the IP sectors, say the five
highest-ranked sectors in table 5.2 Let X M,t represent the vector of output
2
IP numbers are typically released with a 1-month lag, revised up to 3 months after
their initial release, and further subject to an annual revision. Both to confirm initial
releases and to independently track economic activity, the Institute for Supply Management
constructs an index of manufacturing production based on nationwide surveys. In addition,
several Federal Reserve Banks including Dallas, Kansas City, New York, Philadelphia, and
Richmond produce similar indices that are meant to capture real-time changes in activity
at a more regional level. Of course, a central issue pertaining to these surveys is that
gathering information on a large number of sectors in a timely fashion is costly, so the
scope of the surveys is generally limited.
Fig. 5.—Distribution of Ri2(F) of sectoral growth rates. A, 1972–83; B, 1984–2007
sectoral versus aggregate shocks
15
TABLE 6
Information Content of IP Contained in Individual
Sectors: Fraction of Variability of IP Explained by a
Subset of Sectors (Percentage Points)
Number of Sectors
Ordered by Ri2(F)
5
10
20
1972–83
1984–2007
83.2
87.6
96.2
74.1
78.1
82.8
Note.—The entries show Var (wXM,t)/ Var (gt), where XM,t are
the set of M elements of Xt with the largest values of Ri2(F). Results
are shown for M p 5, 10, and 20. The vector w is computed using
¯ Xt, which implies that
the constant share approximation gt ≈ w
¯ , where S is a selection matrix that satisfies
w p (SSXXS )⫺1(SSXXw)
XM,t p SXt . The various average shares, parameters, and moments
are estimated over the periods shown in the column headings.
growth rates associated with these M sectors. The minimum mean square
error linear predictor of g t based on X M,t is w X M,t , where w are the linear
regression coefficients. Table 6 shows the fraction of variability of g t
explained by w X M,t using the five highest-ranked sectors in table 5. Prior
to 1984, these five sectors alone account for 83 percent of the variation
in the growth rate of the aggregate index. In addition, 96 percent of
the variability in IP growth rates is captured by considering only the 20
highest-ranked sectors (out of 117) over 1972–83. This fraction is somewhat smaller (83 percent) over the Great Moderation period. In either
case, however, it is apparent that information about movements in IP
turns out to be concentrated in a small number of sectors. Contrary to
conventional wisdom, these sectors are not necessarily those with the
largest weights, the most volatile output growth, or the most links to
other sectors (e.g., electric power generation). Since aggregate IP is
driven mainly by common shocks, what matters is that those sectors also
move with common shocks.
IV.
Structural Factor Analysis
An important assumption underlying the estimation of factors in the
previous section was that the covariance matrix of the uniquenesses,
u t , in equation (3) satisfies weak cross-sectional dependence. However,
as discussed in Long and Plosser (1983), Horvath (1998), Carvalho
(2007), and elsewhere, sectoral linkages between the industrial sectors
may lead to the propagation of sector-specific shocks throughout the
economy in a way that generates comovement across sectors. In other
words, these linkages may effectively transform shocks that are specific
to particular sectors into common shocks and thereby explain in part
16
journal of political economy
the variability of aggregate output. As discussed in Horvath (1998) and
Dupor (1999), the strength of this amplification mechanism depends
importantly on the structure of production linkages between sectors.
In this section, we use the Bureau of Economic Analysis (BEA) estimates of the input-output and capital use matrices describing U.S. sectoral production to quantify the effects of shock propagation on the
volatility of the aggregate IP index. Because this calculation requires a
model that incorporates linkages between sectors, the first subsection
describes a generalization of the framework introduced in Long and
Plosser (1983) and extended by Horvath (1998). In Horvath’s model,
each sector uses intermediate materials produced in other sectors. However, for analytical tractability, capital in a given sector can be produced
only within that sector and depreciates entirely within the period. In
addition, Horvath abstracts from labor supply considerations. Our generalization relaxes these assumptions by allowing for a conventional
labor supply decision and a standard depreciation rate on capital. More
important, it allows for capital used in a sector to be produced by other
sectors in a manner consistent with BEA estimates of capital use. Therefore, shocks to an individual sector may be disseminated to other sectors
because of interlinkages in materials and capital use and, over time,
because the effects of interlinkages on capital are persistent under a
standard depreciation rate, in a way that potentially contributes to aggregate fluctuations.
This section also bridges the reduced-form factor approach of Section
III with the more structural approaches to sectoral analysis introduced
by Long and Plosser (1983). In particular, it describes the conditions
under which the factor model in (3) may, in fact, be interpreted as the
reduced form of a structural model with explicit sectoral linkages. It
then illustrates how the structural model may be used to filter out the
effects of these linkages and presents quantitative findings based on a
variety of experiments.
A.
A Canonical Model with Sectoral Linkages
Consider an economy composed of N distinct sectors of production
indexed by j p 1, … , N. Each sector produces a quantity Yjt of good j
at date t using capital, K jt , labor, L jt , and materials that are produced
in the other sectors, M ijt , according to the technology
(写 M ) L
N
Yjt p A jt K jtaj
ip1
gij
ijt
N g
1⫺aj ⫺Sip1
ij
jt
,
(4)
where A jt is a productivity index for sector j.
The fact that each sector uses materials from other sectors represents
a source of interconnectedness in the model. An input-output matrix
sectoral versus aggregate shocks
17
for this economy is an N#N matrix G with typical element gij . The
column sums of G give the degree of returns to scale in materials in
each sector. The row sums of G measure the importance of each sector’s
output as materials to all other sectors. Put simply, we can think of the
rows and columns of G as “sell to” and “buy from” shares for each sector,
respectively.
Denote the vector of capital shares by a p (a1 , a 2 , … , aN), let
A t p (A 1t, … , ANt) be the vector of productivity indices, and suppose
that ln (A t) follows the random walk process
ln (A t) p ln (A t⫺1) ⫹ ␧t,
(5)
where ␧t p (␧1t, … , ␧Nt) is a vector-valued martingale difference process
with covariance matrix S␧␧. The degree to which sectoral productivity is
influenced by aggregate shocks will be reflected in the matrix S␧␧. When
S␧␧ is diagonal, sectoral productivity is driven only by idiosyncratic
shocks, whereas common shocks are reflected in nonzero off-diagonal
elements of S␧␧.
In each sector j, the capital stock evolves according to the law of
motion
K jt⫹1 p Z jt ⫹ (1 ⫺ d)K jt,
(6)
where Z jt denotes investment in sector j and d is the depreciation rate.
Investment in sector j is produced using the amount Q ijt of sector i
output by way of the constant returns to scale technology
写
冘
N
Z ji p
ip1
N
Q vijtij ,
vij p 1.
(7)
ip1
The fact that new capital goods in a given sector are produced using
other sectors’ output represents an additional source of interconnectedness in the model. A capital use matrix for this economy is an N #
N matrix V with typical element vij . The row sums of V measure the
importance of each sector’s output as a source of investment to all other
sectors.
A representative agent derives utility from the consumption of these
N goods according to
冘 冘(
⬁
E0
N
bt
tp0
jp1
C jt1⫺j ⫺ 1
⫺ wL jt ,
1⫺j
)
(8)
where the labor specification follows Hansen’s (1985) indivisible labor
model. In addition, each sector is subject to the following resource
constraint:
冘
N
C jt ⫹
ip1
冘
N
M jit ⫹
Q jit p Yjt .
(9)
ip1
The model is closely related to previous work including Long and
18
journal of political economy
Plosser (1983), Horvath (1998), and the benchmark model in Carvalho
(2007), all of which admit analytical solutions for the competitive equilibrium. In online-only Appendix B, we show that the deterministic
steady state of the model continues to be analytically tractable, which
is key for our disaggregation of the data into 117 sectors, and that a
linear approximation of the model’s first-order conditions and resource
constraints around that steady state yields a vector ARMA(1, 1) model
for sectoral output growth, X t p [D ln (Y1t), … , D ln (YNt)]:
(I ⫺ FL)X t p (P 0 ⫹ P 1 L)␧t,
(10)
where L is the lag operator and F, P0, and P1 are N#N matrices that
depend only on the model parameters, a, G, V, b, j, w, and d.
Suppose that innovations to sectoral productivity, ␧t , reflect both aggregate shocks, represented by the k#1 vector St , and idiosyncratic
shocks, represented by the vector vt :
␧t p LSSt ⫹ vt,
(11)
where LS is a matrix of coefficients that determine how the aggregate
disturbances affect productivity in individual sectors. We assume that
(St , vt) is serially uncorrelated, St and vt are mutually uncorrelated, and
Svv p E(vtvt ) is a diagonal matrix (so the idiosyncratic shocks are
uncorrelated).
Given equations (10) and (11), the evolution of sectoral output
growth can then be written as the dynamic factor model
X t p L(L)Ft ⫹ u t,
(12)
where
L(L) p (I ⫺ FL)⫺1(P 0 ⫹ P 1 L)LS,
Ft p St , and
u t p (I ⫺ FL)⫺1(P 0 ⫹ P 1 L)vt.
In other words, the vector of sectoral output growth rates in this multisector extension of the standard growth model produces an approximate factor model as a reduced form. The common factors in this
reduced form, Ft , are associated with aggregate shocks to sectoral productivity, St . The reduced-form idiosyncratic shocks, u t , reflect linear
combinations of the underlying structural sector-specific shocks, vt . Importantly, even though the elements of vt are uncorrelated, sectoral
linkages will induce some cross-sectional dependence among the elements of u t by way of the matrix (I ⫺ FL)⫺1(P 0 ⫹ P 1 L). This dependence
may in turn lead the reduced-form factor model to overestimate the
importance of aggregate shocks. Said differently, by ignoring the covariance in u t that arises from production linkages, the reduced-form
factor model may incorrectly attribute any resulting comovement in
sectoral output growth to aggregate shocks.
sectoral versus aggregate shocks
19
To eliminate the propagation effects of idiosyncratic shocks induced
by production linkages, we use equation (10) and construct ␧t as a
filtered version of the observed growth rate data X:3
␧t p (P 0 ⫹ P 1 L)⫺1(I ⫺ FL)X t.
(13)
Factor analytic methods can then be applied directly to ␧t to estimate
the relative contribution of aggregate shocks, St , and sector-specific
shocks, vt , for the variability of aggregate output.
B.
Choosing Values for the Model Parameters
We interpret the model as describing the sectoral production indices
analyzed in Sections II and III. Thus, we abstract from output of the
agriculture, public, and service sectors. In order to construct the filtered
series described in equation (13), we must first choose values for the
model’s parameters. A subset of these parameters is standard and is
chosen in accordance with previous work on business cycles: j p 1, w
p 1, b p 0.99, and d p 0.025. The choice of input-output matrix, G,
and capital use matrix, V, requires more discussion.
Our choice of parameter values describing the final goods technology
(gij and ai) derives from estimates of the “use tables” constructed by the
BEA. We use the BEA’s use tables from 1997 (although in our robustness
analysis in Sec. VI we show results using the 1977 use table). The BEA
input use tables measure the value of inputs (given by commodity codes)
used by each industry (given by industry codes). By matching commodity
and industry codes for the 117 industries in our data, we obtain the
value of inputs from each industry used by every other industry. The
use tables also include compensation of employees (wages) and other
value added (rents on capital). A column sum in the use table represents
total payments from a given sector to all other sectors (i.e., material
inputs, labor, and capital) and defines the value of output in that sector.
A row sum in the use table gives the importance of a given sector as an
input supplier to all other sectors, measured as the value of inputs in
other sectors’ production. Hence, input shares, gij , are estimated as
dollar payments from industry j to industry i expressed as a fraction of
the value of production in sector j.
The parameters that describe the production of investment goods
(vij) derive from estimates of capital flow tables provided by the BEA,
again for 1997. The capital flow table shows the destination of new
investment in equipment, software, and structures, in terms of the industries purchasing or leasing the new investment. Similar to the use
3
In order to recover the innovations to productivity, ␧t , from current and lagged values
of Xt , the process must be invertible; i.e., the roots of FP0 ⫹ P1zF must lie outside the unit
circle.
20
journal of political economy
table, the capital flow table provides the most detailed view of investment
by commodity and by industry. By matching commodity and industry
codes, we obtain the value of investment goods from each industry
purchased by all other industries. A column sum in the capital flow
table represents total payments for investment goods from a given industry to all other industries. A row sum in the capital flow table gives
the importance of a given sector as a source of investment to all other
sectors. Thus investment shares, vij , are estimated as dollar payments
from industry j to industry i expressed as a fraction of the total investment expenditures of sector j. Appendix A discusses measurement issues
associated with the input use and capital use tables.
The remaining set of parameters that need to be estimated are those
making up S␧␧, the covariance matrix of the structural productivity
shocks, ␧t . We choose two calibrations for S␧␧ that help highlight the
degree to which the model is able to propagate purely idiosyncratic
shocks and thus effectively transform these shocks into common shocks.
In the first calibration, S␧␧ is a diagonal matrix with entries given by the
sample variance of ␧t in the different IP sectors. The sample variance
of ␧t is computed over different sample periods (e.g., 1972–83, 1984–
2007) to account for heteroskedasticity that may be important for the
Great Moderation. Because this calibration uses uncorrelated sectoral
shocks, it allows us to determine whether sectoral linkages per se can
explain the covariance in sectoral production that is necessary to generate the variability in aggregate output.
In the second calibration, we use a factor model to represent S␧␧.
That is, we model ␧t as shown in equation (11), where St is a k#1 vector
of common factors and vt is an N#1 vector of mutually uncorrelated
sector-specific idiosyncratic shocks. In this model, S␧␧ p LS SSS LS ⫹ Svv ,
where LS and the covariance matrices SSS and Svv are estimated using
the principal components estimator of St constructed from the sample
values of ␧t . This second calibration allows two sources of covariance in
sectoral output: a structural component arising from sectoral linkages
and a statistical component arising from aggregate shocks affecting sectoral productivity.4
C.
Results from the Structural Analysis
Comovement from sectoral linkages.—Table 7 summarizes the results. Columns 1–4 show the average pairwise correlation of sectoral growth rates,
4
We do not attempt to identify the source of the common factors affecting sectoral
productivity. These factors may reflect aggregate shocks affecting all sectors in the economy
or, given that our data comprise only the industrial sector, may reflect idiosyncratic shocks
arising in the service, financial, public, or agricultural sectors. See Carvalho and Gabaix
(2010) for an interesting discussion of the role these sectors have played in the Great
Moderation and the increase in volatility during the 2008–9 recession.
sectoral versus aggregate shocks
21
TABLE 7
Sectoral Correlations and Volatility of IP Growth Rates Implied by the
Structural Model
Model with
Uncorrelated
Shocks
Data
Model with Two
Factors
Sample Period
r̄ij
(1)
jg
(2)
r̄ij
(3)
jg
(4)
r̄ij
(5)
jg
(6)
R 2(S)
(7)
1972–83
1984–2007
.27
.11
8.8
3.6
.05
.04
5.1
3.1
.26
.10
9.5
4.1
.81
.50
Note.—The columns labeled r̄ij show the average pairwise correlation of sectoral growth
rates, the columns labeled jg show the standard deviation of IP growth rates, and the
column labeled R 2(S) shows the fraction of the variability in IP associated with the common
factors S.
x it , and the standard deviation of aggregate IP growth, jg , for the data
and as implied by the model with uncorrelated ␧ shocks (S␧␧ diagonal).
Columns 5–7 show the average pairwise correlation, standard deviation
of aggregate IP growth, and the fraction of IP variability explained by
aggregate shocks in the model, where ␧ depends on two common factors
(S␧␧ p LS SSS LS Svv).
Four results are evident in table 7. First, the model with input-output
linkages alone and uncorrelated sector-specific shocks implies noticeably less comovement across sectors than in U.S. data. For example, in
the 1972–83 sample period, the average pairwise correlation in the data
is 0.27, but the model with uncorrelated shocks implies an average
pairwise correlation of only 0.05. Because the model with uncorrelated
shocks underpredicts the correlation in sectoral growth rates, it is unable
to replicate all of the variability in aggregate IP growth. The second
result is that the model with two common factors closely matches the
data’s sectoral covariability and the variability in aggregate IP (although
the model overpredicts the variability somewhat). Third, although the
model’s internal propagation mechanism with purely idiosyncratic
shocks falls short of reproducing the sectoral comovement seen in the
data, it does not mean that this mechanism is inoperative. Thus, the
fraction of IP variability explained by aggregate shocks is smaller than
that explained by the reduced-form factors in both sample periods. For
example, over 1984–2007, the reduced-form model attributes 87 percent
of the variability in aggregate IP to the common factors, F, whereas the
structural model attributes only 50 percent to the common shocks, S.
Finally, aggregate shocks explain a smaller fraction of the variability in
aggregate IP in the 1984–2007 period than in the 1972–83 period. Column 7 of the table shows that this fraction fell from 0.81 to 0.50. Of
course, this implies that the contribution of idiosyncratic shocks to IP
22
journal of political economy
variability increased from 19 percent in the early period to 50 percent
in the latter period. In Section VI, we list the sectors with the most
important idiosyncratic shocks and discuss how this has changed over
time.
V.
Deconstructing the Empirical Results
The previous literature on multisector models has at times argued that,
despite sectoral linkages, idiosyncratic shocks should average out in
aggregation. Most notably, Dupor (1999) provides a careful analysis of
the conditions under which this averaging out occurs in the framework
studied by Horvath (1998). It is instructive to examine our empirical
conclusions in the context of this previous literature. We do this using
three models: the model in Long and Plosser (1983), the benchmark
model used by Carvalho (2007), and the model used by Horvath (1998)
and Dupor (1999). Each of these models admits an analytic solution
that fits into our generic solution (10).
In the economy of Long and Plosser (1983), households have log
preferences over consumption and leisure, materials are delivered with
a one-period lag, and there is no capital. In that economy, the evolution
of sectoral output growth follows the generic solution (10) with F p
G , P 0 p I, and P 1 p 0:
X t p G X t⫺1 ⫹ ␧t.
(14)
In his benchmark economy, Carvalho (2007) studies a version of our
environment with no capital, the same preferences over leisure, and log
preferences over consumption. In that economy, sectoral output growth
evolves as in (10) with F p 0, P 0 p (I ⫺ G )⫺1, and P 1 p 0:
X t p (I ⫺ G )⫺1␧t .
(15)
Finally, the economic environment in Horvath (1998) and Dupor (1999)
is a version of our economy with sector-specific capital subject to full
depreciation within the period (so that V p I and d p1), no labor, and
log preferences over consumption. In that economy, sectoral output
follows (10) with F p (I ⫺ G )⫺1ad , P 0 p (I ⫺ G )⫺1, and P 1 p 0, where
ad is a diagonal matrix with the values ai on the diagonal:
X t p (I ⫺ G )⫺1adX t⫺1 ⫹ (I ⫺ G )⫺1␧t.
(16)
The solution to each of these models, (14), (15), and (16), yields the
following expressions for the covariance matrix of sectoral growth rates,
respectively:
sectoral versus aggregate shocks
冘
23
⬁
SLong-Plosser
p
XX
ip0
(G )iS␧␧ Gi,
Carvalho
SXX
p (I ⫺ G )⫺1 S␧␧(I ⫺ G)⫺1,
冘
(17)
⬁
SHorvath-Dupor
p
XX
ip0
i
[(I ⫺ G )⫺1ad](I
⫺ G )⫺1 S␧␧(I ⫺ G)⫺1[ad(I ⫺ G)⫺1]i.
In each of the models, off-diagonal elements in the input-output matrix
G propagate sector-specific ␧ shocks to other sectors, although the exact
form of the propagation depends on the specifics of the model. Said
differently, in each of the models, diagonal S␧␧ matrices are transformed
in nondiagonal SXX matrices through input-output linkages. These expressions show that the variance of aggregate output depends on S␧␧,
G, and (in the Horvath-Dupor economy) ad in a fairly complicated and
model-specific way.
Dupor (1999) studies a version of the Horvath (1998) model under
two key restrictions: (i) G has a unit eigenvector, so that Gl p kl, where
l is the unit vector and k is a scalar; and (ii) all capital shares are equal,
so that ad p aI, where a is a scalar. Under these restrictions, it is
possible to derive simple expressions for the variance of the equally
weighted aggregate growth rate, denoted with a superscript ew,
t
g tew p N ⫺1 冘ip1 x it p N ⫺1l X t . In particular, the variance of g tew in each
of the models is given by
jg2ew(Long-Plosser) p (1 ⫺ k 2 )⫺1j¯ ij ,
jg2ew(Carvalho) p (1 ⫺ k)⫺2j¯ ij ,
(18)
jg2ew(Horvath-Dupor) p [(1 ⫺ k ⫺ a)(1 ⫺ k ⫹ a)]⫺1j¯ ij ,
where j̄ij p N ⫺2 冘i 冘j E(␧it␧jt) denotes the value of the average element
of S␧␧.5 When idiosyncratic shocks are uncorrelated so that S␧␧ is diagonal, j̄ij denotes the average variance of sectoral shocks divided by N.
Moreover, under the restrictions on G and ad described by restrictions
i and ii, the variance of the aggregate growth rate is proportional to
j̄ij not only in Horvath (1998) but in all the models, where the details
of the model give rise to different factors of proportionality.
The expressions in (18) lead to three conclusions. First, if the idiosyncratic errors are uncorrelated (or weakly correlated), so that
limNr⬁ j¯ ij p 0, then limNr⬁ jg2ew p 0. This is Dupor’s well-known result on
the irrelevance of idiosyncratic shocks for aggregate volatility applied
5
⫺1 When Gl p kl , ad p al , and gew
l Xt , eqq. (14)–(16) imply simple representations
t p N
ew
¯ t p N⫺1l ␧t . Specifically, gew
¯ t for the Long-Plosser econfor gew
t as a function of ␧
t p kgt⫺1 ⫹ ␧
ew
⫺1
⫺1
ew
⫺1
¯ t for
omy, gt p (1 ⫺ k) ␧¯ t for the Carvalho economy, and gew
t p (1 ⫺ k) agt⫺1 ⫹ (1 ⫺ k) ␧
the Horvath-Dupor economy. The expressions given in (18) follow directly from these
representations.
24
journal of political economy
to all three models. The second result is that S␧␧ affects jg2ew only through
the value of its average element, j̄ij . Thus, for example, severe heteroskedasticity associated with a handful of sectors being subject to large
shocks affects jg2ew only through its effect on the average element in S␧␧.
This effect will be small when the number of sectors is large. Finally,
when ␧ depends on common shocks S, the fraction of the variability in
jg2ew associated with the common shocks (denoted R 2(S) in the last section) is the fraction of j̄ij associated with S, which is the same for each
of the models.
All of these conclusions rest on the restriction that Gl p kl . In other
words, the elements in each row of the input-output matrix sum to the
same constant. Recall that the ith row sum of G represents the intensity
of sector i as a material provider to the economy. Therefore, the row
sum restriction implies that all sectors are equally important in this
regard. Figure 6A shows the row sums for the input-output matrix for
IP. Evidently, in contrast to restriction i above, the row sums of the inputoutput matrix for IP differ considerably across sectors. In particular, the
“skewness” of the row sums is consistent with the notion emphasized by
Carvalho (2007) that a few sectors play a key role as input providers.
This suggests that the conclusions that follow from Dupor’s equal-rowsum restriction may not be relevant for the IP data, or more to the
point, they must be investigated by evaluating the general expressions
in (17) and their generalizations for our model, which is done in table
8.
Table 8 summarizes the implied variability in aggregate IP and sectoral
covariability for various versions of the model. It investigates the role
of the input-output matrix G, which has been the focus of much of the
previous literature, and also the role of the parameters that govern
sectoral interactions in investment (V and d), which have not been
investigated previously. The analysis is carried out using models in which
S␧␧ is a diagonal matrix (so that all shocks are idiosyncratic) using the
value from the last section, which we denote by SBenchmark
. However,
␧␧
because the various models imply different scaling for the shocks (i.e.,
the diagonal elements of P0 differ in the models), we also show results
Benchmark
for S␧␧ p hS␧␧
, where h is a model-specific scale factor described
below. Thus, scaled comparisons capture only the internal mechanics
of the different models with ␧ shocks that are normalized across models.
The first row of table 8 shows the average correlation of sectoral
growth rates (r̄ij p 0.19 from table 2), the standard deviation of IP
growth rates (jg p 5.8 percent from table 1), and the standard deviation
of IP growth associated with the diagonal elements of SXX (jg(diag) p
1.9 percent from table 3), computed over the full sample (1972–2007)
period. The other entries show the values implied by the various models,
where the entry for jg (the standard deviation of IP growth rates) uses
Fig. 6.—Some characteristics of sectoral production. A, Intensity of sectors as material
N
input providers to other sectors ( jp1 gij ); B, importance of capital in sectoral production
N
( jp1 vij).
冘
冘
26
journal of political economy
TABLE 8
Selected Summary Statistics for Data and Various Models with Uncorrelated
Shocks (1972–2007 Sample Period)
1. Data
2. Benchmark model
3. Long-Plosser
4. Carvalho
5. Horvath-Dupor
6. Benchmark, V p I
7. Benchmark, d p 1
8. Long-Plosser, G average
9. Carvalho, G average
10. Horvath-Dupor, G average, ad p aI
11. Benchmark, G average,
ad p aI
12. Benchmark, S␧␧ p j2I
2
g
2
g,Benchmark
j (scaled)/
(scaled)
(5)
r̄ij
(1)
jg
(2)
jg(diag)
(3)
jg
(scaled)
(4)
.19
.04
.01
.04
.06
.02
.06
.01
.04
5.80
3.87
2.66
3.15
3.76
3.86
3.74
1.61
2.60
1.85
1.88
2.07
1.64
1.81
2.43
1.72
1.39
1.53
3.82
2.38
3.56
3.84
2.94
4.04
2.15
3.15
1.00
.39
.87
1.01
.59
1.12
.32
.68
.05
2.89
1.58
3.40
.79
.05
.04
3.30
5.72
1.71
2.99
3.57
3.55
.87
.86
j
Note.—Entries for rows 2–12 correspond to models that are described in the text. The
term r̄ij denotes the average pairwise correlation of sectoral growth rates, jg denotes the
standard deviation of IP growth rates, jg(diag) denotes the standard deviation computed
using the diagonal elements of SXX (the covariance matrix of sector growth rates),
jg(scaled) denotes the standard deviation with shocks that are scaled so that the model’s
implied value for jg(diag) is equal to the value for the data, and col. 5 shows j2g (scaled)
for each model relative the value in the benchmark model.
Benchmark
Benchmark
S␧␧ p S␧␧
, the entry for jg(scaled) uses S␧␧ p hS␧␧
, and h is
chosen so that the model-implied variance of IP growth associated with
the diagonal elements of SXX, j(diag), corresponds to the value in the
data (which is 1.85 percent from the first row in the table).
The second row of the table shows results for the benchmark model
considered in the last section. As discussed in that section, idiosyncratic
shocks in the benchmark model underpredict the correlation between
sectoral growth rates (the model’s average value is 0.04 compared to
0.19 in the data) and underpredict the variability in the aggregate
growth rate (the model predicts a scaled standard deviation of 3.8 percent compared to 5.8 percent in the data).
Rows 3–5 show results for the Long-Plosser, Carvalho, and HorvathDupor economies described above, solved using the same parameter
values as in our benchmark model. In the Long-Plosser model, materials
are subject to a one-period delivery lag. Therefore, output growth initially reacts to shocks as though there were no spillovers (i.e., the coefficient on ␧t in eq. [14] is I), and input-output linkages propagate
only this initial “neutral” response. Moreover, the Long-Plosser model
does not have capital. Taken together, these features lead to a lower
sectoral versus aggregate shocks
27
correlation among the sectors and a smaller aggregate variance than in
the benchmark model shown in row 2. While Carvalho also abstracts
from capital, materials are used within the quarter so that idiosyncratic
shocks are contemporaneously amplified through propagation to other
sectors (i.e., ␧t is immediately amplified by [I ⫺ G ]⫺1 in eq. [15]). As
shown in row 4, this change increases the sectoral correlation and aggregate variability relative to the Long-Plosser model, although the
scaled aggregate variability is still somewhat smaller than the benchmark
model.
The Horvath-Dupor model adds sector-specific capital (so that V p
I) but imposes full depreciation within the period (so that d p 1),
shown in row 5 of table 8. This model produces results much in line
with the benchmark model. In the Horvath-Dupor environment, not
only are idiosyncratic shocks propagated to other sectors contemporaneously, as in Carvalho, but these shocks are then further propagated
through their effects on capital accumulation, by way of (I ⫺ G )⫺1ad in
(16), where ad is the matrix of capital shares. Therefore, aggregate
variability is more pronounced in Horvath-Dupor than in the models
of either Long-Plosser or Carvalho.
To understand why the Horvath-Dupor model produces results similar
to our benchmark model, rows 6 and 7 parse the two key restrictions
imposed by Horvath-Dupor relative to our benchmark in Section IV.
The first, that V p I, is shown in row 6. This restriction reduces the
covariability (by eliminating a channel in which the sectors interact)
and the variability of aggregate output. Figure 6B shows the row sums
of V and suggests why imposing V p I results in a reduction of correlation and volatility. The distribution of the row sums of V is highly
skewed. Roughly 85 percent of the sectors provide no capital to other
sectors, and only a handful of sectors are responsible for the bulk of
intersector trade in capital goods.6 The effect of the second restriction,
d p 1, is shown in row 7. This restriction increases sectoral covariability,
by reducing the intertemporal smoothing from capital accumulation,
and the variance of output. Thus, the Horvath-Dupor model produces
numerical results similar to those of the benchmark because it eliminates
two features that have countervailing effects.
Rows 8–11 of table 8 modify G and ad so that they (counterfactually)
satisfy the restrictions used by Dupor (1999). In particular, the actual
input-output matrix is replaced by a symmetric version with common
diagonal and common off-diagonal elements, that is, gii p gdiag for all
i and gij p goffdiag for all i ( j, and gdiag and goffdiag were chosen as the
6
Six sectors (commercial and service industry machinery, metalworking machinery,
industrial machinery, computer and peripheral equipment, navigational/measuring/electromedical/control instruments, and automobiles and light duty motor vehicles) account
for 78 percent of the off-diagonal elements in V.
28
journal of political economy
average values in the actual input-output matrix. Similarly, ad was replaced by aI, where a was chosen as the average capital share. Rows 8–
11 show that these restrictions lead to a reduction in the sectoral correlation and aggregate IP volatility relative to the corresponding models
that do not impose these restrictions. This holds not only for the LongPlosser, Carvalho, and Horvath-Dupor models (shown in rows 8–10) but
also for the more general model with capital interactions and realistic
depreciation (shown in row 11).
The final row of the table investigates the effect of heteroskedasticity
on sectoral output. Recall that in the restricted models of this section,
aggregate variability depends only on j̄ij , the value of the average element in S␧␧, and not on the distribution of elements within S␧␧. Therefore, even if a few sectors were to have large shocks, this will have only
a small effect on the aggregate variance because the number of sectors
is large. However, this argument does not necessarily hold in the unrestricted model of Section IV. In row 12, the model is solved using
S␧␧ p j 2I, where j 2 is chosen as the average sample variance (so that
j̄ij is unchanged). The result is a slight reduction in covariability and
(scaled) variance of the aggregate relative to the benchmark model
shown in row 2. Put another way, the fact that some sectors are subject
to larger idiosyncratic shocks than other sectors leads to a somewhat
larger variance in the aggregate series (3.82 in row 2 vs. 3.55 in row
12).
Column 5 of the table shows the aggregate volatility for each of the
models relative to the values in the benchmark model. The entries range
from 0.32 (for the Long-Plosser model using the counterfactual value
of the input-output matrix) to 1.12 percent (for the benchmark model
with full depreciation). This highlights the fact that, in the context of
the Long-Plosser model and its descendants, the importance of idiosyncratic shocks can vary widely and depends critically on details of the
specification and the values of parameters that govern intra- and intertemporal linkages.
VI.
Level of Aggregation and Time Variation in the Input-Output
Matrix
This section addresses the robustness of our findings along two dimensions. First, we assess how our findings are affected by the level of sectoral
disaggregation. Second, we explore whether changes in the structure
of sectoral linkages in U.S. production over time have changed the
propagation of idiosyncratic shocks.
The results summarized thus far have used IP disaggregated to (essentially) the four-digit level. As discussed above, this yields 117 sectors
and is the finest level of disaggregation that allows us to compute the
sectoral versus aggregate shocks
29
TABLE 9
Selected Summary Statistics with Different Levels of Sectoral Aggregation
1972–83
1984–2007
r̄ij
Two-digit level
(26 sectors)
Three-digit level
(88 sectors)
Four-digit level
(117 sectors)
r̄ij
Data
(1)
Model with
Diagonal S␧␧
(2)
Data
(4)
Model with
Diagonal S␧␧
(5)
R 2(S)
(3)
R 2(S)
(6)
.38
.09
.76
.22
.07
.53
.29
.05
.85
.13
.05
.53
.27
.05
.81
.11
.04
.50
Note.—The table shows the average pairwise correlation of sectoral growth rates (r̄ij)
for different levels of sectoral disaggregation for the data and for the model with uncorrelated shocks (S␧␧ a diagonal matrix). The columns labeled R 2(S) show the fraction of
variability in aggregate IP associated with two common factors (S) computed at the level
of aggregation shown.
input-output and capital use matrices G and V. Table 9 compares results
from models estimated at coarser levels of aggregation (88 three-digit
and 26 two-digit sectors). The table shows that the average pairwise
correlation of sectoral growth rates (r̄ij) increases as the level of aggregation becomes more coarse but that the main conclusions for the
model estimated using the four-digit data continue to hold for the threedigit and two-digit data. In particular, the models with uncorrelated
shocks are unable to capture all the covariability in sectoral growth rates
that is estimated from the data. Because of this, these models underpredict the variability in aggregate IP. That said, the fraction of variability
associated with aggregate shocks, R 2(S), is remarkably stable across the
different levels of aggregation. In particular, the conclusions that the
post-1983 decline in the aggregate volatility is associated with a decline
in the variability of the common factors and that roughly 50 percent of
the variability of IP was associated with sector-specific shocks during the
1984–2007 period are robust to the level of disaggregation.
Another important issue involves the evolution of sectoral production
linkages in the U.S. economy over the 1972–2007 sample period. To
investigate the effect of these changes before and after the Great Moderation, we consider two BEA benchmark years, 1977 and 1997. BEA
benchmark input use tables are available only every 5 years. However,
the use tables for the years prior to 1997 are not updated by the BEA
to reflect the reclassification of industries by NAICS definitions and are
broken down instead according to Standard Industrial Classification
(SIC) codes. For consistency, therefore, we extend our analysis to vintage
IP data provided by the Board of Governors, where sectors are disag-
30
journal of political economy
TABLE 10
Comparing Results (Model with V p I) Sectoral Correlations and Volatility of
IP Growth Rates Implied by Structural Model
Data
Model
with
Uncorrelated
Shocks
Structural
Model with
Two Factors
Reduced-Form
Model with
Two Factors
Sample Period
G
(1)
r̄ij
(2)
jg
(3)
r̄ij
(4)
jg
(5)
R 2(S)
(6)
R 2(F)
(7)
1972–83
1984–2007
1967–83
1984–2002
G1997
G1997
G1977
G1977
.27
.11
.21
.10
8.8
3.6
8.5
3.9
.02
.02
.03
.02
3.7
2.2
4.0
2.4
.88
.69
.83
.73
.89
.87
.85
.94
Note.—The columns labeled r̄ij show the average pairwise correlation of sectoral growth
rates, the columns labeled jg shows the standard deviation of IP growth rates, the column
labeled R 2(S) shows the fraction of the variability of IP associated with aggregate shocks
S, and R 2(F) shows the variability of IP associated with the reduced-form factors, F. Results
are shown using the 1997 input-output matrix G1997 and the sectors classified by NAICS
and for the 1977 matrix G1977 and the sectors classified by SIC.
gregated by SIC codes, and that cover the period 1967–2002. Unfortunately, capital flow tables broken down according to SIC codes were
never constructed for the level of disaggregation considered here.
Therefore, in this exercise, we are limited to the model with V p I.
Table 10 summarizes results for models estimated over pre- and post1983 periods using the 1997 input-output matrix (G1997) and sectors
classified by NAICS and using the 1977 input-output matrix (G1977) and
sectors classified by SIC. The table shows the average pairwise correlation
of sectoral growth rates (r̄ij) computed from the data along with the
standard deviation of IP growth rates (jg). The corresponding values
are shown for the version of the model driven by sector-specific shocks
(i.e., using a diagonal S␧␧ matrix). Also shown are the fraction of variability of aggregate variability associated with the common factors from
the structural model (R 2(S)) and from the reduced-form model
(R 2(F )). Examination of the table entries suggests that our key conclusions are unaffected by changes in the input-output matrix.
Finally, while the empirical results suggest that much of the variability
in IP is associated with the aggregate shocks, S, sector-specific shocks
explain 20 percent of the variability in IP during the 1972–83 period
and 50 percent of its variability in the 1984–2007 period. Table 11 lists
the 10 sectors that are most responsible for this variability in each of
the sample periods computed using the models with two factors. To
account for changes in the structure of U.S. production, panel A shows
results for the idiosyncratic shocks constructed using the 1977 inputoutput matrix for the period 1967–83 (using the SIC classification) and
TABLE 11
Fraction of Variability of IP Explained by Sector-Specific Shocks
Rank
Sector
Fraction
A. 1972–83 SIC (V p I)
1
2
3
4
5
6
7
8
9
10
Basic steel and mill products
Coal mining
Motor vehicles, trucks, and buses
Utilities
Oil and gas extraction
Copper ores
Iron and other ores
Petroleum refining and miscellaneous
Motor vehicle parts
Electronic components
.064
.034
.008
.007
.005
.004
.003
.003
.004
.002
B. 1984–2007 NAICS (V p I)
1
2
3
4
5
6
7
8
9
10
Iron and steel products
Electric power generation, transmission, and distribution
Semiconductors and other electronic components
Oil and gas extraction
Automobiles and light duty motor vehicles
Organic chemicals
Aerospace products and parts
Motor vehicle parts
Natural gas distribution
Support activities for mining
.042
.036
.026
.017
.017
.017
.015
.013
.012
.011
C. 1984–2007 NAICS (V p V1997)
1
2
3
4
5
6
7
8
9
10
Iron and steel products
Semiconductors and other electronic components
Electric power generation, transmission, and distribution
Oil and gas extraction
Organic chemicals
Automobiles and light duty motor vehicles
Natural gas distribution
Motor vehicle parts
Support activities for mining
Other basic inorganic chemicals
.078
.076
.046
.031
.026
.024
.020
.018
.014
.013
Note.—The entries show the fraction of variability of IP growth rates explained by
sector-specific shocks (uit ) from the estimated two-factor models. Results are shown for
the 10 sectors with the largest fractions. Results in panel A are for the model estimated
using the 1977 value of the input-output matrix (G1977), sectors classified by SIC, and with
V p I. Results in panel B are for the model estimated using the 1997 value of the inputoutput matrix (G1997), sectors classified by NAICS, and with V p I. Results in panel C use
the NAICS sectors with 1997 values of input-output and capital use matrices (G1997 and
V1997).
32
journal of political economy
panel B uses the 1997 input-output matrix over 1984–2007 (using the
NAICS classification). As in table 11, because the capital flow table is
not available for the SIC data, these results are calculated using the
model with V p I. Panel C shows the results corresponding to panel B
but using V1997 constructed from the BEA’s 1997 capital use table.
We make three comments about the sectors highlighted in the table.
First, in a comparison of panels A and B, the highest-ranked sectors
tend to be those that serve as inputs to many other sectors. This is the
case, for example, of basic steel and mill products in the pre-1984 period
(6.4 percent), which corresponds to iron and steel products after 1984
(4.2 percent). It is also true of utilities prior to 1984, which corresponds
approximately to electric power generation and distribution in the second sample period. Second, in a comparison of panels B and C, the
ranking of industries in the two panels is very similar (nine industries
appear in both panels and the top six industries coincide), although in
panel C the fraction of variability of IP associated with the sector-specific
shocks is higher. Thus, while ignoring production linkages through capital investment affects the estimate of the fraction of aggregate variability
attributed to idiosyncratic shocks (biasing it down because of the neglected linkages), it does not affect the ranking of the most important
sector-specific shocks. Finally, table 11 points to some changes in the
structure of U.S. production. For example, coal mining, a traditional
industry ranked second prior to 1984 (at 3.4 percent), is no longer
among the 10 highest-ranked sectors in the second sample period.
VII.
Conclusions
In this paper, we explore various leading explanations underlying the
volatility of industrial production and its decline since 1984. We find
that neither time variation in the sectoral shares of IP nor their distribution is an important factor in determining the variability of the aggregate IP index. Instead, the analysis reveals that aggregate shocks
largely explain changes in aggregate IP, and a decrease in the volatility
of these shocks explains why aggregate IP is considerably less variable
after 1984. Because of this decline in the variability of aggregate shocks,
the relative importance of sector-specific shocks has increased substantially over the Great Moderation period. When we take into account
sectoral linkages that characterize U.S. production, in both materials
and investment, sector-specific shocks explain approximately 20 percent
of the variation in IP growth prior to 1984 and 50 percent of IP fluctuations after the onset of the Great Moderation. We also provide evidence that changes in the structure of the input-output matrix between
1977 and 1997 have not led to a more pronounced propagation of
sectoral shocks.
sectoral versus aggregate shocks
33
The analysis also highlights the conditions under which a class of
neoclassical multisector growth models first studied by Long and Plosser
(1983) admit an approximate factor model as a reduced form. In doing
so, it bridges two literatures: one that has relied on factor analytic methods to assess the relative importance of aggregate and idiosyncratic
shocks and the other rooted in more structural calibrated models that
explicitly take into account sectoral linkages in production. In the
reduced-form factor model, aggregate shocks emerge as the common
output factors. The idiosyncratic shocks in the reduced form are associated with sector-specific shocks, but because of sectoral linkages, these
can be cross-sectionally correlated.
Appendix A
Data on industrial production are obtained from the Board of Governors of the
Federal Reserve System and disaggregated according to the NAICS. The raw
data are indices of real output and sectoral shares. While data on the vast majority
of sectors are directly available from the Board of Governors, data are missing
for some sectors, and those missing are approximated using the board’s recommended methodology. For example, if industry C is composed of industry A
and industry B, then the growth rate of C’s output is approximated by
IPCt
wAt⫺1(IPAt/IPAt⫺1) ⫹ wBt⫺1(IPBt/IPBt⫺1)
p
,
IPCt⫺1
wAt⫺1 ⫹ wBt⫺1
where wit is the share of industry i at date t, and wCt p wAt ⫹ wBt . Alternatively,
if industry C is made up of industry A less industry B, then
IPCt
w (IP /IP ) ⫺ wBt⫺1(IPBt/IPBt⫺1)
p At⫺1 At At⫺1
IPCt⫺1
wAt⫺1 ⫺ wBt⫺1
and wCt p wAt ⫺ wBt . As mentioned in the text, we also make use of vintage IP
data, provided by the Board of Governors, which are disaggregated according
to SIC codes. Growth rates and shares for missing sectors are computed in the
manner we have just described. A summary of the NAICS IP data is provided
in table A1.
Benchmark input-output tables, available every 5 years, are obtained from the
BEA. The use table measures the dollar value of inputs given by commodity
codes, used by each industry given by industry codes, as well as payments to
other factors such as labor and capital. Commodity codes are matched to industries to provide a matrix showing the dollar value of inputs purchased by
each industry from each industry. We consider benchmark tables for 1977 and
1997, which are broken down according SIC and NAICS industry definitions,
respectively.
In matching the input-output matrices with IP data, we find the smallest set
of common industries for which we can match the IP data to the input-output
data. Because the two data sources are originally disaggregated to different levels,
the resulting sectoral breakdown represents collections of either NAICS or SIC
industry levels (depending on whether we are using current or vintage IP data).
TABLE A1
117 Sectors, NAICS Industry Classification
Standard Deviation
of Quarterly
Growth Rates
Sector
Logging
Oil and gas extraction
Coal mining
Iron ore mining
Gold, silver, and other ore mining
Copper, nickel, lead, and zinc mining
Nonmetallic mineral mining and quarrying
Support activities for mining
Electric power generation, transmission, and distribution
Natural gas distribution
Animal food
Grain and oilseed milling
Sugar and confectionery products
Fruit and vegetable preserving and specialty foods
Dairy products except frozen
Ice cream and frozen desserts
Animal slaughtering and processing
Seafood product preparation and packaging
Bakeries and tortilla
Coffee and tea
Other food except coffee and tea
Soft drinks and ice
Breweries
Wineries and distilleries
Tobacco
Fiber, yarn, and thread mills
Fabric mills
Textile and fabric finishing and fabric coating
mills
Carpet and rug mills
Curtain and linen mills
Other textile product mills
Apparel
Leather and allied products
Sawmills and wood preservation
Veneer, plywood, and engineered wood products
Millwork
Wood containers and pallets
All other wood products
Pulp mills
Paper and paperboard mills
Paperboard containers
Paper bags and coated and treated paper
Other converted paper products
Printing and related support activities
Petroleum refineries
34
Weight
1972–83
1984–2007
.27
5.79
1.06
.09
.16
.15
.63
1.06
17.9
3.8
79.7
132.0
30.3
61.4
14.3
20.6
14.2
5.7
15.4
37.9
22.0
14.2
10.4
28.6
7.65
1.65
.41
.77
.55
1.03
.72
.11
1.29
.14
1.23
.18
.95
.60
.45
.26
1.07
.22
.69
8.0
16.0
8.1
12.4
23.4
9.6
4.8
9.7
10.4
28.6
4.3
39.0
11.1
7.5
14.7
26.6
12.6
23.6
16.5
6.9
16.7
10.3
8.9
12.0
11.3
7.4
12.8
6.3
24.6
4.2
17.7
7.1
8.9
9.1
20.5
19.3
15.3
9.9
.31
.20
.16
.20
1.83
.34
.44
.33
.34
.08
.29
.08
1.59
.71
.39
.37
2.30
1.66
17.5
26.9
18.3
16.4
10.8
12.3
25.8
26.9
21.1
12.2
26.8
16.7
16.0
15.7
11.4
15.1
7.2
13.9
11.6
16.3
12.6
8.7
8.9
13.2
11.3
12.1
9.0
8.7
15.9
10.8
6.2
6.6
10.0
8.3
4.9
7.6
TABLE A1 (Continued)
Standard Deviation
of Quarterly
Growth Rates
Sector
Paving, roofing, and other petroleum and coal
products
Organic chemicals
Industrial gas
Synthetic dyes and pigments
Other basic inorganic chemicals
Resins and synthetic rubber
Artificial and synthetic fibers and filaments
Pesticides, fertilizers, and other agricultural chemicals
Pharmaceuticals and medicines
Paints and coatings
Adhesives
Soap, cleaning compounds, and toilet preparation
Other chemical product and preparation
Plastics products
Tires
Rubber products except tires
Pottery, ceramics, and plumbing fixtures
Clay building materials and refractories
Glass and glass products
Cement
Concrete and products
Lime and gypsum products
Other nonmetallic mineral products
Iron and steel products
Alumina and aluminum production and processing
Nonferrous metal smelting and refining (except
aluminum)
Copper and nonferrous metal rolling, drawing, extruding, and alloying
Foundries
Fabricated metals: forging and stamping
Fabricated metals: cutlery and handtools
Architectural and structural metal products
Boiler, tank, and shipping containers
Fabricated metals: hardware
Fabricated metals: spring and wire products
Machine shops; turned products; and screws, nuts,
and bolts
Coating, engraving, heat treating, and allied activities
Other fabricated metal products
Agricultural implements
Construction machinery
Mining and oil and gas field machinery
Industrial machinery
Commercial and service industry machines/other
general purpose machines
35
Weight
1972–83
1984–2007
.33
1.39
.21
.15
.57
.79
.34
17.8
19.0
19.9
30.9
26.7
32.9
38.4
11.3
12.8
17.7
16.8
21.7
11.4
12.9
.49
2.52
.40
.13
1.42
.94
2.27
.44
.40
.12
.16
.63
.19
.68
.11
.37
1.70
12.9
5.6
18.3
16.7
9.8
10.4
16.8
44.9
20.0
14.7
22.7
11.7
28.0
13.6
19.1
16.3
41.5
10.1
6.7
13.7
11.2
9.3
9.6
6.0
14.0
7.8
12.5
16.9
6.9
16.8
9.2
17.1
8.2
19.5
.52
17.6
13.4
.13
50.6
19.0
.35
.79
.50
.35
1.15
.59
.29
.20
41.2
19.5
16.7
15.0
11.7
9.1
18.4
16.5
17.4
9.1
8.0
8.0
6.0
6.6
9.5
8.8
1.05
15.5
8.8
.40
1.33
.50
.44
.29
.74
11.6
10.9
22.0
46.7
28.4
11.9
9.4
5.8
27.6
32.0
21.9
20.5
2.19
11.8
6.2
36
journal of political economy
TABLE A1 (Continued)
Standard Deviation
of Quarterly
Growth Rates
Sector
Ventilation, heating, air conditioning, and commercial refrigeration equipment
Metalworking machinery
Engine, turbine, and power transmission equipment
Computer and peripheral equipment
Communications equipment
Audio and video equipment
Semiconductors and other electronic components
Navigational/measuring/electromedical/control
instruments
Magnetic and optical media
Electric lighting equipment
Small electrical household appliances
Major electrical household appliances
Electrical equipment
Batteries
Communication and energy wires and cables
Other electrical equipment
Automobiles and light duty motor vehicles
Heavy duty trucks
Motor vehicle bodies and trailers
Motor vehicle parts
Aerospace products and parts
Railroad rolling stock
Ship and boat building
Other transportation equipment
Household and institutional furniture and kitchen
cabinets
Office and other furniture
Medical equipment and supplies
Other miscellaneous manufacturing
Newspaper publishers
Periodical, book, and other publishers
Weight
1972–83
1984–2007
.72
.86
25.0
19.0
19.2
9.7
.80
1.52
1.55
.19
2.34
20.2
17.4
10.7
37.7
18.8
14.9
16.4
16.6
43.4
16.5
2.33
.20
.34
.15
.37
.89
.16
.21
.47
2.32
.16
.41
3.09
3.17
.23
.51
.16
10.7
28.0
15.2
19.4
36.3
17.0
22.0
19.1
20.7
51.2
50.3
29.9
28.2
14.1
30.2
11.3
25.4
6.2
21.6
8.0
18.4
15.4
8.7
16.6
14.8
8.0
24.8
35.3
17.1
13.3
13.4
22.4
10.8
17.1
.87
.62
1.18
1.36
1.47
1.99
18.5
13.4
6.7
11.3
6.0
9.0
7.1
8.7
5.4
5.8
6.7
7.3
Note.—The column labeled “weight” shows the sector’s percentage point weight in
aggregate IP. Standard deviations of quarterly growth rates are in percentage points at an
annual rate.
The approach is as follows: taking the finest available partition of industries
from the benchmark input-output tables, we match as many industries as possible; the remaining industries with no matches are aggregated until a match is
found. The results are collections of industries whose level of disaggregation
ranges from two-digit to four-digit levels. Results reported in the text are for the
highest level of disaggregation available, the four-digit level, unless otherwise
stated.
Data from the 1997 input-output table are matched with IP data disaggregated
sectoral versus aggregate shocks
37
using NAICS definitions over the period 1972Q1–2007Q4. Similarly, data from
the 1977 input-output table are matched with IP data broken down by SIC code
from 1967Q1 to 2002Q3. The reclassification of industries from the SIC system
to the NAICS system, and the fact that older input-output tables are not updated
according to NAICS definitions, makes the use of vintage IP data necessary since
there is no easy mapping from SIC to NAICS definitions.
As a practical matter, the distinction between what constitutes materials, as
captured by G, and investment goods, as captured by V, is not always unambiguous. The BEA distinguishes between materials and capital goods by estimating
the service life of different commodities. Thus, commodities expected to be
used in production within the year are defined as materials.
There are two notable measurement issues that arise with the construction
of the capital flow table. First, the table accounts for the purchases of new capital
goods and does not include the purchases of used assets. This means, for example, that a firm’s purchases of used trucks in a given industry from another
industry will not be recorded as investment in the capital flow table even when
the trucks’ remaining service life is well in excess of a year.7 Second, as discussed
in McGrattan and Schmitz (1999), maintenance expenditures are largely absent
from the table. The absence of data on purchases of used capital goods leads
us to ignore the first measurement problem. However, we do adjust the capital
flow table for maintenance expenditures. McGrattan and Schmitz provide evidence that in aggregate manufacturing, maintenance and repair payments may
be as large as 50 percent of total investment. If sector j’s missing expenditures
are proportional to the capital expenditures to each sector i reported in the
capital flow table, then the missing expenditures will have no effect on the
shares vij . However, there is reason to suspect that this is not the case. Specifically,
much maintenance and repair take place using within-sector resources, and yet
many of the diagonal elements of the capital flow table are very small or zero.
It is difficult to know exactly what to do about this bias. We have taken a roughand-ready approach and added 25 percent of the reported total expenditures
from the table to the diagonal. Essentially, this assumes that the published table
underestimates total expenditures by roughly 50 percent, that half of this is
associated with within-sector expenditures on maintenance and repair, and that
the other half is proportional to capital expenditures reported in the table. In
our numerical experiments, we found that some of our results were sensitive to
this adjustment. In particular, adjusting the diagonal elements by values much
less than 25 percent (say, using 15 percent) led to a noninvertible model, so
that ␧t could not be recovered from current lagged values of X t . However, using
larger values closer to that of McGrattan and Schmitz (say, using 35 percent)
led to results much like those obtained using 25 percent.
7
The difficulty of the problem lies in having to keep track of the entire age distribution
of the different types of capital that switch industries. It is not an issue for the overall
measurement of investment in IP since the capital is always within IP but can matter for
tracking investment at a more disaggregated level.
38
journal of political economy
References
Anderson, T. W. 1984. An Introduction to Multivariate Statistical Analysis. 2nd ed.
New York: Wiley.
Bai, Jushan. 2003. “Inferential Theory for Factor Models of Large Dimensions.”
Econometrica 71:135–71.
Bai, Jushan, and Serena Ng. 2002. “Determining the Number of Factors in
Approximate Factor Models.” Econometrica 70:191–221.
Carvalho, Vasco M. 2007. “Aggregate Fluctuations and the Network Structure
of Intersectoral Trade.” Manuscript, Univ. Chicago.
Carvalho, Vasco M., and Xavier Gabaix. 2010. “The Great Diversification and
Its Undoing.” Manuscript, Stern School Bus., New York Univ.
Chamberlain, Gary, and Michael Rothschild. 1983. “Arbitrage, Factor Structure,
and Mean-Variance Analysis of Large Asset Markets.” Econometrica 51:1281–
1304.
Connor, Gregory, and Robert A. Korajczyk. 1986. “Performance Measurement
with the Arbitrage Pricing Theory.” J. Financial Econ. 15:373–94.
Dupor, Bill. 1999. “Aggregation and Irrelevance in Multi-sector Models.” J. Monetary Econ. 43:391–409.
Forni, Mario, Marc Hallin, Marco Lippi, and Lucrezia Reichlin. 2000. “The
Generalized Factor Model: Identification and Estimation.” Rev. Econ. and Statis.
82:540–54.
Forni, Mario, and Lucrezia Reichlin. 1998. “Let’s Get Real: A Dynamic Factor
Analytical Approach to Disaggregated Business Cycle.” Rev. Econ. Studies 65:
453–74.
Gabaix, Xavier. 2011. “The Granular Origins of Aggregate Fluctuations.” Econometrica, forthcoming
Hansen, Gary. 1985. “Indivisible Labor and the Business Cycle.” J. Monetary Econ.
16:309–37.
Horvath, Michael. 1998.“Cyclicality and Sectoral Linkages: Aggregate Fluctuations from Independent Sectoral Shocks.” Rev. Econ. Dynamics 1:781–808.
———. 2000. “Sectoral Shocks and Aggregate Fluctuations.” J. Monetary Econ.
45:69–106.
Long, John B., Jr., and Charles I. Plosser. 1983. “Real Business Cycles.” J.P.E. 91:
39–69.
———. 1987. “Sectoral vs. Aggregate Shocks in the Business Cycle.” A.E.R. 77:
333–36.
McGrattan, Ellen, and James Schmitz. 1999. “Maintenance and Repair: Too Big
to Ignore.” Fed. Reserve Bank Minneapolis Q. Rev. 23:2–13.
Quah, Danny, and Thomas J. Sargent. 1993. “A Dynamic Index Model for Large
Cross Sections.” In Business Cycles, Indicators, and Forecasting, edited by James
H. Stock and Mark W. Watson. Chicago: Univ. Chicago Press (for NBER).
Shea, John. 2002. “Complementarities and Comovements.” J. Money, Credit, and
Banking 34:412–33.
Stock, James H., and Mark W. Watson. 2000. “Forecasting Using Principal Components from a Large Number of Predictors.” J. American Statis. Assoc. 97:
1167–79.